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Unformatted text preview: Solution to HW#4 EGM 6812 Fall 2009 1. Using the differential conservation of momentum equation as a starting point, derive the vorticity transport equation, ( 29 2 D V Dt ϖ ϖ ν ϖ = ⋅ ∇ + ∇ r r r r , for a flow with constant density and viscosity. Hints: start by taking 2 b DV f p V Dt ρ ρ μ ∇ × = ∇ + ∇ r r r . Solution: (note both the underline and upper arrow denote vector) Note: b f ρ ∇ × = r assuming b f r is only due to gravitational force OR b f r is conservative p ∇ ×∇ ≡ 2 2 V μ μ ϖ ∇ ×∇ = ∇ ur r ( ) DV V V Dt t ϖ ρ ρ ∂ ∇ × = + ∇× ⋅∇ ∂ ur r r r Since 1 ( ) ( ) 2 V V V V V V ⋅∇ = ∇ ⋅ × ∇× r r r r r r => ( 29 ( 29 V V V ϖ ∇ × ⋅∇ = ∇ × × ur r r r Since ∇ x (A x B ) = A ∇⋅ B A ⋅∇ B B ∇⋅ A + B ⋅∇ A , we have ( 29 V V V V V ϖ ϖ ϖ ϖ ϖ ∇ × × = ∇ ⋅ ⋅∇ ∇ ⋅ + ⋅∇ ur ur ur ur ur r r r r r Thus, the LHS of the equation becomes...
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This note was uploaded on 09/09/2010 for the course EGM 6812 taught by Professor Renweimei during the Fall '09 term at University of Florida.
 Fall '09
 RENWEIMEI

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